'They do punish false accusers when appropriate'.
That depends on your definition of appropriate. They do, maybe, sometimes, when the accuser has admitted lying. When the accusation is so outlandish or perverse that the world and his wife, the judge and the prosecutor are sure the accuser is lying, and the evidence contradicts itself, but the accuser doesn't admit it, they don't, and the accusation stays on the record indefinitely. It stays on the record when the accusation is so frivolous that it isn't actually a crime, but the prosecutor decided to go to trial 'as a matter of policy'. Do you see any of the complainants in any of the cases that have fallen apart recently, and where the accuser has clearly been lying, (you know, 'thanks for the great sex, honey' in one breath and 'he raped me' in the next, or texts to friends saying 'he didn't rape me, you know,) being prosecuted? Jemma Beale had to accuse men in double figures before anything was done to stop her.
People who insist on saying that if you're falsely accused 'truth will out and all will be well' every time are as deluded as Alison Saunders who insists that she doesn't think that anyone wrongly convicted is in jail. Not long ago a policeman was falsely accused, I forget which force at the moment, but he was investigated for a whole 5 weeks before the case was closed and he needed 6 MONTHS off work because of the stress it caused him. How much stress does it cause falsely accused men who are investigated, or at least told they are being investigated, for 2-3 years?
People who say that 'it's no big deal' if you're accused falsely or that being falsely accused is so rare that throwing those so accused to the wolves is a small price to pay for those who need encouragement to make complaints, have no idea and need to do some research, not throw unsubstantiated statistics around like confetti.